Permutation Polynomials Of Degree Research Articles

Boomerang uniformity of normalized permutation polynomials of low degree

Differential uniformity of permutation polynomials has been studied intensively in recent years due to the differential cryptanalysis of S-boxes. The boomerang attack is a variant of differential cryptanalysis which combines two differentials for the upper part and the lower part of the block cipher. The boomerang uniformity measures the resistance of block ciphers to the boomerang attack. In this paper, by using the resultant elimination method, we study the boomerang uniformity of normalized permutation polynomials of the low degree over finite fields. As a result, we determine the boomerang uniformity of all normalized permutation polynomials of degree up to six over the finite field $${\mathbb {F}}_{q}$$.

Permutation polynomials of degree 8 over finite fields of characteristic 2

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree 8 over F2r with r>3. By Bartoli et al. (2017) [1], a polynomial f of degree 8 over F2r is exceptional if and only if f−f(0) is a linearized PP, which has already been classified. So it suffices to search for non-exceptional PPs of degree 8 over F2r, which exist only when r⩽9 by a previous result. This can be exhausted by the SageMath software running on a personal computer. To facilitate the computation, some requirements after linear transformations and explicit equations by Hermite's criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP f of degree 8 over F2r (with r>3) exists if and only if r∈{4,5,6}, and such f is explicitly listed up to linear transformations.

Lengths for Which Fourth Degree PP Interleavers Lead to Weaker Performances Compared to Quadratic and Cubic PP Interleavers.

In this paper, we obtain upper bounds on the minimum distance for turbo codes using fourth degree permutation polynomial (4-PP) interleavers of a specific interleaver length and classical turbo codes of nominal 1/3 coding rate, with two recursive systematic convolutional component codes with generator matrix . The interleaver lengths are of the form or , where is a product of different prime numbers greater than three. Some coefficient restrictions are applied when for a prime , condition is fulfilled. Two upper bounds are obtained for different classes of 4-PP coefficients. For a 4-PP , , the upper bound of 28 is obtained when the coefficient of the equivalent 4-permutation polynomials (PPs) fulfills or when and , , for any values of the other coefficients. The upper bound of 36 is obtained when the coefficient of the equivalent 4-PPs fulfills and , , for any values of the other coefficients. Thus, the task of finding out good 4-PP interleavers of the previous mentioned lengths is highly facilitated by this result because of the small range required for coefficients and . It was also proven, by means of nonlinearity degree, that for the considered inteleaver lengths, cubic PPs and quadratic PPs with optimum minimum distances lead to better error rate performances compared to 4-PPs with optimum minimum distances.

A classification of permutation polynomials of degree 7 over finite fields

Up to linear transformations, a classification of all permutation polynomials (PPs) of degree 7 over Fq is obtained for any odd prime power q>7. Since all exceptional polynomials of degree 7 are precisely known up to linear transformations, it suffices to work on non-exceptional PPs of degree 7, which exist only when q⩽409 by our previous result. This can be exhausted by the SageMath software running on a personal computer. To facilitate the computation, some requirements after linear transformations and equations by Hermite's criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP f of degree 7 exists over a finite field Fq of odd order q>7 if and only if q∈{9,11,13,17,19,23,25,27,31,49}, and f is explicitly listed up to linear transformations.

Parallel access by butterfly networks for any degree permutation polynomial and ARP interleavers

Using more processors for parallel turbo decoding is an important issue to speed up the processing at the receiver of a communication system. Butterfly networks used to map the addresses of extrinsic values represent an elegant and simple solution in parallel turbo decoding. Recently, it has been shown that quadratic permutation polynomial (QPP) interleavers allow an easy way to compute the control bits for a butterfly network. In this paper we show that not only QPP interleavers, but any degree permutation polynomial (PP) interleavers and almost regular permutation (ARP) interleavers also allow the same easy way to compute the control bits required in butterfly networks. As a consequence, it is useful to apply the butterfly networks in parallel turbo decoding when using these performant algebraic interleavers.

Correction to: The number of different true permutation polynomial based interleavers under Zhao and Fan sufficient conditions

In the original version of this article, unfortunately, there are mistakes in some formulas for determining the number of true different cubic, fourth degree, and fifth degree permutation polynomial based interleavers under Zhao and Fan sufficient conditions.

On the difference between permutation polynomials

The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if p>(d2−3d+4)2, then there is no complete mapping polynomial f in Fp[x] of degree d≥2. For arbitrary finite fields Fq, a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of f.Cohen, Mullen and Shiue generalized the Chowla–Zassenhaus–Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f+g are both permutation polynomials of degree d≥2 over Fp, with p>(d2−3d+4)2, then the degree k of g satisfies k≥3d/5, unless g is constant. In this article, assuming f and f+g are permutation polynomials in Fq[x], we give lower bounds for the Carlitz rank of f in terms of q and k. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials f of Carlitz rank n≥1 that if f+xk is a permutation over Fq, with gcd⁡(k+1,q−1)=1, then k≥(q−n)/(n+3).

Ambiguity, deficiency and differential spectrum of normalized permutation polynomials over finite fields

We obtain exact formulas for the differential spectrum, deficiency and ambiguity of all normalized permutation polynomials of degree up to six over finite fields.

On the number of permutation polynomials over a finite field

In order to extend the results of [Formula: see text] in [P. Das, The number of permutation polynomials of a given degree over a finite field, Finite Fields Appl. 8(4) (2002) 478–490], where [Formula: see text] is a prime, to arbitrary finite fields [Formula: see text], we find a formula for the number of permutation polynomials of degree [Formula: see text] over a finite field [Formula: see text], which has [Formula: see text] elements, in terms of the permanent of a matrix. We write down an expression for the number of permutation polynomials of degree [Formula: see text] over a finite field [Formula: see text], using the permanent of a matrix whose entries are [Formula: see text]th roots of unity and using this we obtain a nontrivial bound for the number. Finally, we provide a formula for the number of permutation polynomials of degree [Formula: see text] less than [Formula: see text].

Permutation polynomials of degree 6 or 7 over finite fields of characteristic 2

In Dickson (1896–1897) [2], the author listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.

The Number of Permutation Polynomials of a Given Degree Over a Finite Field

We relate the number of permutation polynomials in Fq[x] of degree d≤q−2 to the solutions (x1,x2,…,xq) of a system of linear equations over Fq, with the added restriction that xi≠0 and xi≠xj whenever i≠j. Using this we find an expression for the number of permutation polynomials of degree p−2 in Fp[x] in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity. This leads to nontrivial bounds for the number of such permutation polynomials. We provide numerical examples to illustrate our method and indicate how our results can be generalised to polynomials of other degrees.

The Number of Permutation Polynomials of a Given Degree Over a Finite Field

We relate the number of permutation polynomials in F q [ x] of degree d≤ q−2 to the solutions ( x 1, x 2,…, x q) of a system of linear equations over F q , with the added restriction that x i ≠0 and x i ≠ x j whenever i≠ j. Using this we find an expression for the number of permutation polynomials of degree p−2 in F p [ x] in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity. This leads to nontrivial bounds for the number of such permutation polynomials. We provide numerical examples to illustrate our method and indicate how our results can be generalised to polynomials of other degrees.

On a Conjecture of Carlitz

A conjecture of Carlitz on permutation polynomials is as follow: Given an even positive integer n, there is a constant Cn, such that if Fq is a finite field of odd order q with q > Cn, then there are no permutation polynomials of degree n over Fq. The conjecture is a well-known problem in this area. It is easily proved if n is a power of 2. The only other cases in which solutions have been published are n = 6 (Dickson [5]) and n = 10 (Hayes [7]); see Lidl [11], Lausch and Nobauer [9], and Lidl and Niederreiter [10] for remarks on this problem. In this paper, we prove that the Carlitz conjecture is true if n = 12 or n = 14, and give an equivalent version of the conjecture in terms of exceptional polynomials.